Rocky Kolb, Fermilab & Chicago Texas A&M April 2006 Rocky Kolb, Fermilab & Chicago Texas A&M April 2006 All work is the result of collaborations with Sabino Matarrese and Antonio Riotto (Padova) [and occasionally Alessio Notari (McGill)]

 CDM

Do we “know” there is dark energy? Assume model cosmology: – Friedmann model: H 2  k/a 2 =  G  /  – Energy (and pressure) content:    M   R    +  – Input or integrate over cosmological parameters: H , etc. Calculate observables d L (z), d A (z),  Compare to observations Model cosmology fits with  , but not without   All evidence for dark energy is indirect : observed H(z) is not described by H(z) calculated from the Einstein-de Sitter model [spatially flat ( k  from CMB ; matter dominated (  M )]

Take sides! Can’t hide from the data –  CDM too good to ignore – SNIa – Subtraction:  – Age – Large - scale structure – … Dark energy (modify right-hand side of Einstein equations) – “Just” , a cosmological constant – If not constant, what drives dynamics (scalar field) Gravity (modify left-hand side of Einstein equations) – Beyond Einstein (non - GR: branes, etc.) – (Just) Einstein (GR: back reaction of inhomogeneities) H(z) not given by Einstein–de Sitter  H   k /a     G  MATTER  G     G T  (matter)

Modifying the left-hand side Braneworld modifies Friedmann equation Phenomenological approach Gravitational force law modified at large distance Tired gravitons Gravity repulsive at distance R  Gpc n = 1 KK graviton mode very light, m  ( Gpc )  Einstein & Hilbert got it wrong Backreaction of inhomogeneities Freese & Lewis Five-dimensional at cosmic distances Deffayet, Dvali & Gabadadze Gravitons metastable - leak into bulk Gregory, Rubakov & Sibiryakov; Dvali, Gabadadze & Porrati Kogan, Mouslopoulos, Papazoglou, Ross & Santiago Csaki, Erlich, Hollowood & Terning Räsänen; Kolb, Matarrese, Notari & Riotto; Notari; Kolb, Matarrese & Riotto Binetruy, Deffayet, Langlois Carroll, Duvvuri, Turner, Trodden

Acceleration from inhomogeneities Most conservative approach — nothing new – no new fields (like   eV mass scalars) – no extra long-range forces – no modification of general relativity – no modification of gravity at large distances – no Lorentz violation – no extra dimensions, bulks, branes, etc. – no faith-based (anthropic/landscape) reasoning Magnitude?: calculable from observables related to  Why now?: acceleration triggered by era of non-linear structure

Acceleration from inhomogeneities Homogeneous modelInhomogeneous model We think not!

Acceleration from inhomogeneities View scale factor as zero-momentum mode of gravitational field In homogeneous/isotropic model it is the only degree of freedom Inhomogeneities: non-zero modes of gravitational field Non-zero modes interact with and modify zero-momentum mode cosmology scalar-field theory zero-mode a h   i (vev of a scalar field) non-zero modes inhomogeneities thermal/finite-density bkgd. modify a ( t ) modify h  (t) i e.g., acceleration e.g., phase transitions Cosmology  scalar field theory analogue physical effect

Different approaches Expansion rate of an inhomogeneous Universe  expansion rate of homogeneous Universe with   h  i Inhomogeneities modify zero-mode [effective scale factor is a D  V D  ] Effective scale factor has a (global) effect on observables Potentially can account for acceleration without dark energy or modified GR Model an inhomogeneous Universe as a homogeneous Universe model with   h  i Zero mode [ a(t ) / V  ] is the zero mode of a homogeneous model with   h  i Inhomogeneities only have a local effect on observables Cannot account for observed acceleration Standard approach Our approach

Acceleration from inhomogeneities We do not use super-Hubble modes for acceleration. We do not depend on large gravitational potentials such as black holes and neutron stars. We assert that the back reaction should be calculated in a frame comoving with the matter—other frames can give spurious results. We demonstrate large corrections in the gradient expansion, but the gradient expansion technique can not be used for the final answer—so we have indications (not proof) of a large effect. The basic idea is that small-scale inhomogeneities “renormalize” the large-scale properties.

Inhomogeneities–cosmologyInhomogeneities–cosmology Our Universe is inhomogeneous Can define an average density   The expansion rate of an inhomogeneous universe of average density   is NOT! the same as the expansion rate of a homogeneous universe of average density   ! Difference is a new term that enters an effective Friedmann equation — the new term need not satisfy energy conditions! We deduce dark energy because we are comparing to the wrong model universe (i.e., a homogeneous/isotropic model) Ellis, Barausse, Buchert, Ellis, Kolb, Matarrese, Notari, Räsänen, Riotto, Schwarz

Inhomogeneities–exampleInhomogeneities–example ( a  a )  is not  G     Perturbed Friedmann–Lemaître–Robertson–Walker model: Kolb, Matarrese, Notari & Riotto ( a  a is not even the expansion rate) Could   G   play the role of dark energy? 

Inhomogeneities–cosmologyInhomogeneities–cosmology For a general fluid, four velocity u   ( ,  ) (local observer comoving with energy flow) For irrotational dust, work in synchronous and comoving gauge Velocity gradient tensor  is the volume-expansion factor and  i j is the shear (shear will have to be small) For flat FLRW, h ij (t)  a 2 (t)  ij   3H and  i j = 

Local deceleration parameter positive: However must coarse-grain over some finite domain: Evolution and smoothing do not commute: Inhomogeneities and acceleration Buchert & Ellis; Kolb, Matarrese & Riotto although can’t accelerate, can! Hirata & Seljak; Flanagan; Giovannini; Alnes, Amarzguioui & Gron

Define a coarse-grained scale factor: Coarse-grained Hubble rate: Effective evolution equations: not described by a simple p  w  Inhomogeneities and smoothing Kinematical back reaction: Kolb, Matarrese & Riotto astro-ph/ ; Buchert & Ellis

Inhomogeneities and smoothing Kinematical back reaction: For acceleration: Integrability condition (GR): Acceleration is a pure GR effect: – curvature vanishes in Newtonian limit – Q D will be exactly a pure boundary term, and small Particular solution:  Q D  h R i D  const. – i.e.,  eff  Q D ( so Q D acts as a cosmological constant)

InhomogeneitiesInhomogeneities Does this have anything to do with our universe? Have to go to non-perturbative limit! How to relate observables ( d L (z), d A (z), H(z),  ) to Q D &  R  D ? Can one have large effect and isotropic expansion/acceleration? (i.e., will the shear be small?) What about gravitational instability? Toy model proof of principle: Tolman-Bondi dust model Nambu & Tanimoto; Moffat; Tomita, …

Observational consequences Tomita, 2001 Spherical model Overall Einstein–de Sitter Inner underdense 200 Mpc region Compensating high-density shell Calculate d L (z) Compare to SNIa data Fit with  !

CommentsComments “Do you believe?” is not the relevant question Acceleration of the Universe is important; this must be explored How it could go badly wrong: – Backreaction should not be calculated in – frame comoving with matter flow – Series re-sums to something harmless – No reason to stop at first large term – Synchronous gauge is tricky  Residual gauge artifacts  Synchronous gauge develops coordinate  singularities at late time (shell crossings) Problem could be done in Poisson gauge

ConclusionsConclusions Must properly smooth inhomogeneous Universe In principle, acceleration possible even if “locally”  p >  Super-Hubble modes, of and by themselves, cannot accelerate Sub-Hubble modes have large terms in gradient expansion – Newtonian terms can be large but combine as surface terms – Post-Newtonian terms are not surface terms, but small – Mixed Newtonian £ Post-Newtonian terms can be large – Effect from “mildly” non-linear scales The first large term yields effective cosmological constant No reason to stop at first large term Can have w  ? Advantages to scenario: – No new physics – “Why now” due to onset of non-linear era

Many issues: non-perturbative nature shell crossing comparison to observed LSS gauge/frame choices physical meaning of coarse graining Program: can inhomogeneities change effective zero mode? how does (does it?) affect observables? can one design an inhomogeneous universe that accelerates? could it lead to an apparent dark energy? can it be reached via evolution from usual initial conditions? does it at all resemble our universe? large perturbative terms resum to something harmless?

Rocky Kolb, Fermilab & Chicago Texas A&M April 2006 Rocky Kolb, Fermilab & Chicago Texas A&M April 2006 All work is the result of collaborations with Sabino Matarrese and Antonio Riotto (Padova) [and occasionally Alessio Notari (McGill)]